29 August, 2010

Decision by Objectives (DBO) classes applications of numbers into a hierarchy of four different types, each building on the properties of the previous one.

The concepts are not unfamiliar at all to a mathematician, but they are approaching from a management direction rather than from pure mathematics, and I think that is an interesting different perspective.

The first application of numbers is nominal. In this application, numbers are used to name things. DBO gives examples from horse racing: The number on a jockey or a horse is a nominal use. It identifies a particular jockey or horse; but it is meaningless to (for example) add two jockey numbers together, or divide one by another. In Maths, that means it is possible to permute the numbers in any way without losing any information. The numbers, ℕ, are used purely as a set of distinct objects without structure or relation. We could equally well use any other sufficiently large set of symbols (such as letters).

The next application of numbers is ordinal. In this application, numbers are used to order things. In the horse racing example, the finishing order of the horses is an ordinal application. Some horse finishes first, another finishes second, yet another finishes third. In Maths, there is a linear order between the numbers, and we have a relation < or ≤ that makes sense. We can compare numbers in this application, with a notion of bigger or faster or stronger). But it still doesn't make sense to (for example) add or subtract those numbers, or to make statements like "horse X finished 3 places ahead of horse y".

The third application of numbers is interval. It makes sense to subtract interval numbers from each other. The authors are a little vague in DSO about whether they think it makes sense to add interval numbers. I think in some applications it does (how much chocolate did you eat? how much chocolate did I eat? we can compute: how much chocolate did we eat together?) but in some it does not (what is the time now? what time did the meeting start? we can compute: how long has the meeting been going on (subtraction), but adding the time now and the meeting start time doesn't make sense). When it does make sense to add, the numbers form a group with operation +. When it only makes sense to take the difference, then the numbers form something like a torsor - a group without identity.

The final application is ratio. These is use of numbers where ratios make sense - for example, a horse might finish in half the time of another, meaning that the ratio of the time of the first horse to the second horse is 1/2. It makes sense to multiply and divide these numbers; and ratios can also be used to scale interval numbers. Mathematically, we have a (meaningful use of a) field, ℝ or ℚ.

The point of them making these distinctions in DSO was to point out that just because you have some information in your decision making represented as numbers, it doesn't always make sense to do things like taking the mean, or summing, or whatever.